# Class 11 RD Sharma Solutions – Chapter 31 Derivatives – Exercise 31.3

**Question 1. Find the component statements of the following compound statements:**

**(i) The sky is blue, and the grass is green.**

**(ii) The earth is round, or the sun is cold.**

**(iii) All rational numbers are real, and all real numbers are complex.**

**(iv) 25 is a multiple of 5 and 8.**

**Solution:**

**Every compound statement has components .**

(i)The component’s of the statement are:A: The sky is blue.

B: The grass is green.

(ii)The component’s of the statement are:A: The earth is round.

B: The sun is cold.

(iii)The component’s statement are:A: All rational number are real.

B: All real number are complex.

(iv)The component’s of the statement are:A: 25 is multiple of 5.

B: 25 is multiple of 8.

**Question 2. For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer.**

**(i) Students can take Hindi or Sanskrit as their third language.**

**(ii) To entry a country, you need a passport or a voter registration card.**

**(iii) A lady gives birth to a baby boy or a baby girl.**

**(iv) To apply for a driving license, you should have a ration card or a passport.**

**Solution:**

(i)A choice is given to a student to choose either Hindi or Sanskrit as their third language. The student can’t choose both. Therefore, an exclusive “OR” is used because a student cannot take both Hindi and Sanskrit.

(ii)A passport or voter registration card is needed by a person to enter a country. However, a person may have both. Therefore, an inclusive “OR” is used because a person can have both a passport and a voter registration card to enter a country.

(iii)A lady cannot give birth to a baby who is both a boy and a girl, therefore, an exclusive “OR” is used because either a boy or girl is born.

(iv)A ration card or passport is needed to apply a driving license. However, a person may have both.Therefore, an inclusive “OR” is used because a person can have both a ration card and passport to apply for a driving license.

**Question 3. Write the component statements of the following compound statements and check whether the compound statement is true or false:**

**(i) To enter into a public library children need an identification card from the school or a letter from the school authorities.**

**(ii) All rational numbers are real and all real numbers are not complex.**

**(iii) Square of an integer is positive or negative.**

**(iv) x = 2 and x = 3 are the roots of the equation 3x ^{2} – x – 10 = 0.**

**(v) The sand heats up quickly in the sun and does not cool down fast at night.**

**Solution:**

(i)The components are:P: To get into a public library children need an identity card from the school.

Q: To get into a public library children need a letter from the school authorities.

Both of the statements P and Q are true. Therefore, the compound statement is true.

(ii)The components are:P: All rational numbers are real.

Q: All real numbers are not complex.

The statement P is true and Q is false therefore, P and Q both together are False. Therefore, the compound statement is False

(iii)The components are:P: Square of an integer is positive.

Q: Square of an integer is negative.

Both statements P and Q are true. Therefore, the compound statement is True.

(iv)The components of the compound statement are:P: x = 2 is a root of the equation 3x

^{2}– x – 10 = 0Q: x = 3 is a root of the equation 3x

^{2}– x – 10 = 0The statement P is true, but Q is false then P and Q both combined are False. Therefore, the compound statement is False.

(v)The components of the compound statement are:P: The sand heats up quickly in the sun.

Q: The sand does not cool down fast at night.

Both statements P and Q both are False. Therefore, the compound statement is False.

**Question 4. Determine whether the following compound statements are true or false:**

**(i) Delhi is in India and 2 + 2 = 4**

**(ii) Delhi is in England and 2 + 2 = 4**

**(iii) Delhi is in India and 2 + 2 = 5**

**(iv) Delhi is in England and 2 + 2 = 5**

**Solution:**

(i)The components are:A: Delhi is in India.

B: 2 + 2 = 4

Both statements A and B are true. Hence, the compound statement is True.

(ii)The components are:A: Delhi is in England.

B: 2 + 2 = 4

The statement A is false, and B is true. So, both A and B together are false. Hence, the compound statement is False.

(iii)The components are:A: Delhi is in India.

B: 2 + 2 = 5

The statement B is false, and A is true. So, both A and B together are false. Hence, the compound statement is False.

(iv)The components are:A: Delhi is in England.

B: 2 + 2 = 5

Both the statements A and B are false. Hence, the compound statement is False.